Delay of exchange of stabilities in singularly perturbed parabolic problems

N. N. Nefedov ; K. R. Schneider

Trudy Instituta matematiki i mehaniki, Asymptotic expansions, approximation theory, topology, Tome 9 (2003) no. 1, pp. 121-130

Bibliographie

[1] Benoit E., Dynamic Bifurcation, Lect. Notes Math., 1493, Springer-Verlag, Berlin etc., 1991, 219 pp. | MR

[2] Butuzov V. F., “Singularly perturbed parabolic equations in case of intersecting roots of the degenerate equation”, Russ. J. Math. Physics, 9 (2002), 50–59 | MR | Zbl

[3] Butuzov V. F., Nefedov H. H., “Singulyarno vozmuschennaya kraevaya zadacha dlya uravneniya vtorogo poryadka v sluchae smeny ustoichivosti”, Mat. zametki, 63:3 (1998), 354–362 | MR | Zbl

[4] Butuzov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed boundary value problems in case of exchange of stabilities”, J. Math. Anal. Appl., 229 (1999), 543–562 | DOI | MR | Zbl

[5] Butuzov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed reaction-diffusion systems in cases of exchange of stabilities”, Nat. Resour. Model., 13 (2000), 247–269 | MR | Zbl

[6] Butuzov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed elliptic problems in the case of exchange of stabilities”, J. Differ. Equations, 169 (2001), 373–395 | DOI | MR | Zbl

[7] Butuzov V. F., Nefedov H. H., Shnaider K. P., “O singulyarno vozmuschennoi sisteme parabolicheskikh uravnenii v sluchae peresecheniya kornei vyrozhdennogo uravneniya”, Zhurn. vychisl. matematiki i mat. fiziki, 42:2 (2002), 185–196 | MR | Zbl

[8] Butuzov V. F., Nefedov N. N., Schneider K.R., “Singularly perturbed partly dissipative reactiondiffusion systems in case of exchange of stabilities”, J. Math. Anal. Appl., 240 (2002), 543–562 | MR

[9] Butuzov V. F., Smurov I., “Initial boundary value problem for a singularly perturbed parabolic equation in case of exchange of stability”, J. Math. Anal. Appl., 234 (1999), 183–192 | DOI | MR | Zbl

[10] Dumortier F., Smits B., “Transition time analysis in singularly perturbed boundary value problems”, Trans. Amer. Math. Soc., 347 (1995), 4129–4145 | DOI | MR | Zbl

[11] Dumortier F., Roussarie R., Canard cycles and center manifolds, Mem. Amer. Math. Soc., 577, AMS, Providence, 1996, 100 pp. | MR

[12] Haberman R., “Slowly varying jump and transition phenomena associated with algebraic bifurcation problems”, SIAM J. Appl. Math., 37 (1979), 69–106 | DOI | MR | Zbl

[13] Karimov S., “Asimptotika reshenii odnogo klassa differentsialnykh uravnenii s malym parametrom pri starshei proizvodnoi v sluchae smeny ustoichivosti tochki pokoya v ploskosti bystrykh dvizhenii”, Differents. uravneniya, 21 (1985), 1698–1701 | MR | Zbl

[14] Kolesov A. Yu., Rozov N. Kh., ““Okhota na utok” v issledovanii singulyarno vozmuschennykh kraevykh zadach”, Differents. uravneniya, 35 (1999), 1356–1365 | MR | Zbl

[15] Krupa M., Szmolyan P., “Relaxation oscillation and canard explosion”, J. Differ. Equat., 174 (2001), 312–368 | DOI | MR | Zbl

[16] Krupa M., Szmolyan P., “Extending geometric singular perturbation theory to non-hyperbolic points fold and canard points in two dimensions”, SIAM J. Math. Anal., 33 (2001), 286–314 | DOI | MR | Zbl

[17] Lebovitz N. R., Schaar R. J., “Exchange of stabilities in autonomous systems”, Stud. Appl. Math., 54 (1975), 229–260 | MR | Zbl

[18] Lebovitz N. R., Schaar R. J., “Exchange of stabilities in autonomous systems-II. Vertical bifurcation”, Stud. Appl. Math., 56 (1977), 1–50 | MR | Zbl

[19] Nefedov N. N., Schneider K. R., Delayed exchange of stabilities in singularly perturbed systems, prepr. No 270, Weierstrass Inst. Appl. Math. and Stochastics, Berlin, 1996

[20] Nefedov N. N., Schneider K. R., “Immediate exchange of stabilities in singularly perturbed systems”, Different. and Integr. Equat., 12 (1999), 583–599 | MR | Zbl

[21] Nefedov N. N., Schneider K. R., Schuppert A., Jumping behavior in singularly perturbed systems modeling bimolecular reactions, Prepr. No 137, Weierstrafi-Institut für Angewandte Analysis und Stochastik, Berlin, 1994

[22] Neishtadt A. I., “O zaderzhke poteri ustoichivosti pri dinamicheskikh bifurkatsiyakh I”, Differents. uravneniya, 23 (1987), 2060–2067 | MR

[23] Neishtadt A. I., “O zaderzhke poteri ustoichivosti pri dinamicheskikh bifurkatsiyakh II”, Differents. uravneniya, 24 (1988), 226–233 | MR

[24] Pao C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York–London, 1992, 777 pp. | MR

[25] Kolesov A. Yu., Rozov N. Kh., “Problema “buridanova osla” v relaksatsionnykh sistemakh s odnoi medlennoi peremennoi”, Mat. zametki, 65:1 (1999), 153–156 | MR | Zbl

[26] Shishkova M. A., “Issledovanie sistemy differentsialnykh uravnenii s malym parametrom pri starshikh proizvodnykh”, Dokl. AN SSSR, 209 (1973), 576–579 | Zbl

[27] Shchepakina E. A., Sobolev V. A., “Integral manifolds, canards and black swans”, Nonlinear Anal.: Theory, Methods, Appl., 44 (2001), 897–908 | DOI | MR | Zbl

[28] Sobolev V. A., Schepakina E. A., “Integralnye poverkhnosti traektorii-utok s izmenyayuscheisya ustoichivostyu”, Izv. RAEN. Ser. MMMIU, 1:3 (1997), 151–175 | MR | Zbl

[29] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973, 272 pp. | MR

[30] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. shk., M., 1990, 208 pp. | MR

[31] Vasil'eva A. B., Butuzov V. F., Kalachev L. V., The Boundary Function Method for Singular Perturbation Problems, SIAM Stud. Appl. Math., 14, SIAM, Philadelphia, 1995, 221 pp. | MR